Housing Assistance for Veterans Act of 2013 or HAVEN Act - Directs the Secretary of Housing and Urban Development (HUD) to establish a pilot program to award grants to nonprofit organizations that primarily serve veterans or low-income individuals. Requires such grants to be used to rehabilitate and modify the primary residence of disabled or low-income veterans (at a specified limited or no cost to such veterans). Limits grant amounts to $1 million per organization. Requires the Secretary to direct the oversight of grant fund use. Requires a minimum of 50% matching funds by participating organizations.

Requires the Inspector General of HUD to review and report to the Chairmen of specified congressional committees on: (1) the use of appropriated funds by HUD and by grantees under the pilot program, and (2) oversight and accountability of such grantees.